Report from Dagstuhl Seminar 11071
Theory and Applications of Graph Searching Problems
(GRASTA 2011)
Edited by
Fedor V. Fomin1 , Pierre Fraigniaud2 , Stephan Kreutzer3 , and
Dimitrios M. Thilikos4
1
2
3
4
University of Bergen, NO, [email protected]
Université Paris Sud, FR, [email protected]
University of Oxford, GB, [email protected]
National and Kapodistrian University of Athens, GR, [email protected]
Abstract
From February 14, 2012 to February 18, 2012, the Dagstuhl Seminar 11071 “Theory and Applications of Graph Searching Problems (GRASTA 2011)” was held in Schloss Dagstuhl – Leibniz
Center for Informatics. During the seminar, participants presented their current research, and
ongoing work and open problems were discussed. Abstracts of the presentations given during the
seminar as well as abstracts of seminar results and open problems are put together in this paper.
The first section describes the seminar topics and goals in general. The second section contains
the abstracts of the talks and the third section includes the open problems presented during the
seminar.
Seminar 14.–18. February, 2011 – www.dagstuhl.de/11071
1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, G.2.1 Combinatorics, G.2.2 Graph Theory, G.2.3 Applications, I.2.9 Robotics
Keywords and phrases Graph Searching, Pursuit Evasion Games, Cop and Robers Games, Fugitive Search Games
Digital Object Identifier 10.4230/DagRep.1.2.30
1
Executive Summary
Fedor V. Fomin
Pierre Fraigniaud
Stephan Kreutzer
Dimitrios M. Thilikos
License
Creative Commons BY-NC-ND 3.0 Unported license
© Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer, and Dimitrios M. Thilikos
Graph searching is often referred to, in a more playful language, as a pursuit-evasion game
(or, alternatively, cops and robbers game). This is a kind of game where one part is a set of
escaping mobile entities, called evaders (or fugitives), that hide in a graph representing a
network, and the other part is a number of chasing agents, called searchers (or pursuers),
that move systematically in the graph. The game may vary significantly according to the
capabilities of the evaders and the pursuers in terms of relative speed, sensor capabilities,
visibility, etc. The objective of the game is to capture the evaders in an optimal way, where
the notion of optimality itself admits several interpretations.
Except where otherwise noted, content of this report is licensed
under a Creative Commons BY-NC-ND 3.0 Unported license
Theory and Applications of Graph Searching Problems, Dagstuhl Reports, Vol. 1, Issue 2, pp. 30–46
Editors: Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer, and Dimitrios M. Thilikos
Dagstuhl Reports
Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany
Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer, and Dimitrios M. Thilikos
31
Graph searching revealed the need to express in a formal mathematical way intuitive
concepts such as avoidance, surrounding, sense of direction, hiding, persecution, and threatening. There are many variants of graph searching studied in the literature, which are either
application driven, i.e. motivated by problems in practice, or are inspired by foundational
issues in Computer Science, Discrete Mathematics, and Artificial Intelligence including
Information Seeking
Robot motion planning
Graph Theory
Database Theory and Robber and Marshals Games
Logic
Distributed Computing
Models of computation
Network security
The objective of the seminar was to bring researchers from the widest possible variety
of disciplines related to graph searching and we will especially encourage the maximum
interplay between theory and applications. The meeting initiated the exchange of research
results, ideas, open problems and discussion about future avenues in Graph Searching. As a
fruit of this encounter new research results, open problems, and methodologies will appeared,
especially those of interdisciplinary character.
11071
32
11071 – Theory and Applications of Graph Searching Problems (GRASTA 2011)
2
Table of Contents
Executive Summary
Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer, and Dimitrios M. Thilikos . 30
Overview of Talks
Cops and Robbers played on random graphs
Pawel Pralat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
Complexity of Cops and Robber Game
Petr Golovach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
Robotic Pursuit Evasion and Graph Search
Athanasios Kehagias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Polygon reconstruction from local observations
Peter Widmayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
The price of connectivity in graph searching games
Dariusz Dereniowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
On the Fast Edge Searching Problem
Boting Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Algorithms for solving infinite games on graphs
Marcin Jurdzinski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
On the complexity of CSP decompositions
Zoltán Miklos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Searching Games
Maria Serna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
An overview of The Firefighter Problem
Margaret-Ellen Messinger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
30
11
Graphs with average degree smaller than
are burning slowly
Pawel Pralat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
Cops and Robbers on Directed Graphs
Jan Obdrzalek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Cop and robber games when the robber can hide and ride
Nicolas Nisse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Complexity of the cop and robber guarding game
Tomas Valla . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Multi-target ray searching problems
Spyros Angelopoulos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Characterizations of k-cop win graphs
Nancy Clarke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Some thoughts on constrained cops-and-robbers
Gena Hahn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Hypergraph searching as notion justification
Andrei Krokhin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer, and Dimitrios M. Thilikos
33
Monitoring on a Grid
Dieter Mische . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
LIFO-search
Dimitrios M. Thilikos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
Open problems
Cops and Robbers, parameterized algorithms
Fedor V. Fomin and Petr Golovach . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
Computing edge and nodes search numbers on special graph classes
Pinar Heggernes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
Connected node search number
Dimitrios M. Thilikos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Span-width
Isolde Adler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Cop number of toroidal graphs
Gena Hahn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Kelly-width
Paul Hunter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Ray searching
Spyros Angelopoulos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Ratio of monotonicity
Stephan Kreutzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Best strategy to catch the drunk robber
Dimitrios M. Thilikos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
Escaping from random cops
Pierre Fraigniaud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
11071
34
11071 – Theory and Applications of Graph Searching Problems (GRASTA 2011)
3
3.1
Overview of Talks
Cops and Robbers played on random graphs
Pawel Pralat (West Virginia Univ. – Morgantown, US)
License
Creative Commons BY-NC-ND 3.0 Unported license
© Pawel Pralat
We study the vertex pursuit game of Cops and Robbers, in which cops try to capture a
robber on the vertices of the graph. The minimum number of cops required to win on a
given graph G is called the cop number of G. We present asymptotic results for the game of
Cops and Robbers played on random graph. In particular we show that:
the Meyniel’s conjecture holds a.a.s. for a random d-regular graph G(n, d) as well as a
binomial random graph G(n, p) – joint work with Wormald,
the cop number of G(n, p) as a function of an average degree forms an intriguing zigzag
shape – joint work with Luczak,
almost all cop-win graphs contain a universal vertex – joint work with Bonato and
Kemkes.
Other related problems will be mentioned as well.
3.2
Complexity of Cops and Robber Game
Petr Golovach (University of Durham, GB)
License
Creative Commons BY-NC-ND 3.0 Unported license
© Petr Golovach
The Cops and Robbers game was defined independently by Winkler-Nowakowski and Quilliot
in the 1980s and since that time has been studied intensively. Despite of such a study of the
combinatorial properties of the game, almost no algorithmic results on this game are known.
Perhaps the main algorithmic result known about Cops and Robbers game is the observation
that determining whether the cop number of a graph on n vertices is at most k can be done
by a backtracking algorithm which runs in time nO(k) (thus polynomial for fixed k). From
the hardness side, Goldstein and Reingold in 1995 proved that the version of the Cops and
Robbers game on directed graphs is EXPTIME-complete. Also, they have shown that the
version of the game on undirected graphs when the cops and the robber are given their initial
positions is also EXPTIME-complete. They also conjectured that the game on undirected
graphs is also EXPTIME-complete. However, even NP-hardness of the problem was proved
only in 2008 by Fomin, Golovach and Kratochvíl. We survey the known complexity results
about the Cops and Robbers game and its variants and give a list of open problems.
Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer, and Dimitrios M. Thilikos
3.3
35
Robotic Pursuit Evasion and Graph Search
Athanasios Kehagias (Aristotle University of Thessaloniki GR)
License
Creative Commons BY-NC-ND 3.0 Unported license
© Athanasios Kehagias
Robotic Pursuit Evasion (PE) is a hot research area in the robotics community. Among
the various mathematical tools roboticists use to model the PE problem, the Graph Search
(GS) theory is a prominent (but not the only) example. In this talk I will present and
compare several GS-based approaches to robotic PE. I will point out similarities but also
differences between robotic PE and graph search. In particular, I will compare the goals,
methodology and outlook of roboticists, pure mathematicians and applied mathematicians
who have attacked the problem. I will also present some robotic PE problems which require
extensions of the “classical” GS setup and I will briefly discuss models of robotic PE which
use graphs but not the graph search setup.
3.4
Polygon reconstruction from local observations
Peter Widmayer (ETH Zürich, CH)
License
Creative Commons BY-NC-ND 3.0 Unported license
© Peter Widmayer
We study the problem of reconstructing an unknown simple polygon from a series of certain
local observations, similar in spirit to the reconstruction of an unknown network by exploring
it. For mobile agents that move in simple ways inside a polygon, we are interested in
understanding what types of local observations carry enough information to allow polygon
reconstruction. This is part of a more general effort to understand when and how simple
primitives allow mobile agents to draw global conclusions about the environment from local
observations.
3.5
The price of connectivity in graph searching games
Dariusz Dereniowski (Gdansk University of Technology, PL)
License
Creative Commons BY-NC-ND 3.0 Unported license
© Dariusz Dereniowski
In the edge searching problem the goal is to clear a simple graph that is initially entirely
contaminated. The task is performed by a team of searchers that are allowed to make three
types of moves: a searcher is placed on a vertex, a searcher is removed from a vertex, and a
searcher slides from a vertex to one of its neighbors. The fugitive is invisible, fast, and has
complete knowledge about the graph and the strategy used by the searchers. The fugitive is
considered captured if a searcher reaches his location. We are interested in determining the
minimum number of searchers (i.e. the search number) required to search a given graph. In
the connected graph searching problem we have an additional restriction: the subgraph that
is free of the fugitive is always connected. In this talk we discuss the connection between the
search number and the connected search number, including an algorithm that converts a
given search strategy using k searchers into a connected one using at most 2k + 3 searchers.
11071
36
11071 – Theory and Applications of Graph Searching Problems (GRASTA 2011)
3.6
On the Fast Edge Searching Problem
Boting Yang (University of Regina, CA)
License
Creative Commons BY-NC-ND 3.0 Unported license
© Boting Yang
In this talk, we consider the problem of finding the minimum number of steps to capture
the fugitive. We introduce the fast edge searching problem in the edge search model, we
describe relations between the fast edge searching and other searching problems, such as the
fast searching and the node searching problems, and we present some recent progress on
lower bounds and upper bounds of fast search numbers.
3.7
Algorithms for solving infinite games on graphs
Marcin Jurdzinski (University of Warwick, GB)
License
Creative Commons BY-NC-ND 3.0 Unported license
© Marcin Jurdzinski
This talk is a selective survey of algorithms for solving a number of infinite-path-following
games on graphs, such as parity, mean-payoff, and discounted games. The games considered
are zero-sum, perfect-information, and non-stochastic. Several state-of-the-art algorithms for
solving infinite games on graphs are presented, exhibiting disparate algorithmic techniques,
such as divide-and-conquer, dynamic programming/value iteration, local search/strategy
improvement, and mathematical programming, as well as hybrid approaches that dovetail
some of the former. While the problems of solving infinite games on graphs are in NP and
co-NP, and also in PLS and PPAD, and hence unlikely to be complete for any of the four
complexity classes, no polynomial-time algorithms are known for solving them.
3.8
On the complexity of CSP decompositions
Zoltán Miklos (EPFL – Lausanne, CH)
License
Creative Commons BY-NC-ND 3.0 Unported license
© Zoltán Miklos
We give a short overview of the relation of certain graph and hypergraph games (robbers
and cops/marshals) and CSP decompositions. We discuss the complexity of these problems,
in particular the case of tree decompositions. Finally, we report on some progress about the
analogous hypergraph problems.
3.9
Searching Games
Maria Serna (UPC – Barcelona, ES)
License
Creative Commons BY-NC-ND 3.0 Unported license
© Maria Serna
We consider a general multi-agent framework in which a set of n agents are roaming a network
(such as internet or social networks) where m valuable and sharable goods (or resources or
Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer, and Dimitrios M. Thilikos
37
services) are hidden in m different vertices of the network. We analyze several strategic
situations that arise in this setting by means of game theory. To do so we introduce search
games, in those games agents have to select a simple path from a predetermined set of
initial vertices. Depending on how the goods are splitted among the agents we consider
two search game types: finders-share in which the agents that find a good split among
them the corresponding benefit and first-share in which only the agents that first find a
good share the corresponding benefit. We show that finders-share games always have pure
Nash equilibria (PNE). For obtaining this result we introduce the notion of Nash preserving
reduction between strategic games. We show that finders-share search games are Nash
reducible to single-source network congestion games. This is done through a series of Nash
preserving reductions. For first-share search games we show the existence of games with
and without PNE. Furthermore we identify some graph families in which the first-share
search game has always a PNE that is computable in polynomial time. We discuss also some
variants of searching games and the associated graph parameters.
3.10
An overview of The Firefighter Problem
Margaret-Ellen Messinger (Mount Allison University – Sackville, CA)
The Firefighter Problem is a simplified model for the spread of a fire (or disease or computer
virus) in a network. Initially, a fire breaks out at a vertex in a connected graph. At each
subsequent time step, firefighters protect a fixed number of unburned vertices and then the
fire spreads to all unprotected neighbors. Since its introduction in 1995, there has been
a steady growth of both structural and algorithmic results. One possible objective is to
maximize the number of saved vertices: this generally requires a strategy on the part of the
firefighters, while the fire itself spreads without any strategy. Another possible objective
is to find the number of firefighters needed to save a particular number of, or fraction of,
or subset of the vertices. (These objectives are sometimes in conflict.) I will discuss some
interesting results as well as variants and open problems.
3.11
Graphs with average degree smaller than
30
11
are burning slowly
Pawel Pralat (West Virginia Univ. – Morgantown, US)
License
Creative Commons BY-NC-ND 3.0 Unported license
© Pawel Pralat
We consider the following firefighter problem on a finite graph G = (V, E). Suppose that a
fire breaks out at a given vertex v ∈ V . In each subsequent time unit, a firefighter protects
one vertex which is not yet on fire, and then the fire spreads to all unprotected neighbours of
the vertices on fire. Since the graph is finite, at some point each vertex is either on fire or is
protected by the firefighter, and the process is finished. The objective of the firefighter is
to save as many vertices as possible. Let sn(G, v) denote the number of vertices in G the
firefighter can save when a fire breaks out at vertex v ∈ V , assuming the best strategy is
used. The surviving rate ρ(G) of G is defined as the expected percentage of vertices that can
P
be saved when a fire breaks out at a random vertex of G, that is, ρ(G) = n12 v∈V sn(G, v).
The main focus of the talk is on sparse graphs. Let > 0. We show that any graph G
on n vertices with at most ( 15
11 − )n edges can be well protected, that is, ρ(G) > 60 > 0.
11071
38
11071 – Theory and Applications of Graph Searching Problems (GRASTA 2011)
Moreover, a construction of a random graph is proposed to show that the constant
be improved.
3.12
15
11
cannot
Cops and Robbers on Directed Graphs
Jan Obdrzalek (Masaryk University, PL)
License
Creative Commons BY-NC-ND 3.0 Unported license
© Jan Obdrzalek
We survey the current status of cops and robber games on directed graphs. After presenting
the different variants of the game for the most common digraph measures we ask the following
question: Is there a digraph width measure which is powerful (i.e. a big class of problems is
decidable in linear/polynomial time if this measure is bounded), significantly different from
tree-width and yet, at the same time, characterizable by a variant of the cops-and-robber
game for tree-width? We show that, under some standard complexity theory assumption,
this is not so. We also show a new improvement of this result: That we do not need the
measure to be efficiently orientable for our theorem to hold.
3.13
Cop and robber games when the robber can hide and ride
Nicolas Nisse (INRIA Sophia Antipolis, FR)
License
Creative Commons BY-NC-ND 3.0 Unported license
© Nicolas Nisse
In the classical cop and robber game, two players, the cop C and the robber R, move
alternatively along edges of a finite graph G = (V, E). The cop captures the robber if both
players are on the same vertex at the same moment of time. A graph G is called cop win if the
cop always captures the robber after a finite number of steps. Nowakowski, Winkler (1983)
and Quilliot (1983) characterized the cop-win graphs as graphs admitting a dismantling
scheme. In this talk, we characterize in a similar way the cop-win graphs in the game in
which the cop and the robber move at different speeds s0 and s, s0 ≤ s. We also investigate
several dismantling schemes necessary or sufficient for the cop-win graphs in the game in
which the robber is visible only every k moves for a fixed integer k > 1. We characterize the
graphs which are cop-win for any value of k.
3.14
Complexity of the cop and robber guarding game
Tomas Valla (Charles University – Prague, CZ)
License
Creative Commons BY-NC-ND 3.0 Unported license
© Tomas Valla
The guarding game is a game in which several cops has to guard a region in a (directed or
undirected) graph against a robber. The robber and the cops are placed on vertices of the
graph; they take turns in moving to adjacent vertices (or staying), cops inside the guarded
region, the robber on the remaining vertices (the robber-region). The goal of the robber is to
enter the guarded region at a vertex with no cop on it. The problem is to determine whether
Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer, and Dimitrios M. Thilikos
39
for a given graph and given number of cops the cops are able to prevent the robber from
entering the guarded region. The problem is highly nontrivial even for very simple graphs.
It is know that when the robber-region is a tree, the problem is NP-complete, and if the
robber-region is a directed acyclic graph, the problem becomes PSPACE-complete [Fomin,
Golovach, Hall, Mihalák, Vicari, Widmayer: How to Guard a Graph? Algorithmica, DOI:
10.1007/s00453-009-9382-4]. We solve the question asked by Fomin et al. and we show that
if the graph is arbitrary (directed or undirected), the problem becomes ETIME-complete.
3.15
Multi-target ray searching problems
Spyros Angelopoulos (CNRS – Paris, FR)
License
Creative Commons BY-NC-ND 3.0 Unported license
© Spyros Angelopoulos
We consider the problem of exploring m concurrent rays (i.e., branches) using a single searcher.
The rays are disjoint with the exception of a single common point, and in each ray a potential
target may be located. The objective is to design efficient search strategies for locating t
targets (with t ≤ m). This setting generalizes the extensively studied ray search (or star
search) problem, in which the searcher seeks a single target. In addition, it is motivated by
applications such as the interleaved execution of heuristic algorithms, when it is required
that a certain number of heuristics have to successfully terminate. We study the problem
under two different cost measures, and show how to derive optimal search strategies for each
measure.
3.16
Characterizations of k-cop win graphs
Nancy Clarke (Acadia University – Wolfville, CA)
License
Creative Commons BY-NC-ND 3.0 Unported license
© Nancy Clarke
We give two characterizations of the graphs on which k cops have a winning strategy in the
game of Cops and Robber. These generalize the corresponding characterizations that are
known in the one cop case. In particular, we give a relational characterization of k-copwin
graphs, for all finite k, and then use this characterization to obtain a vertex elimination order
characterization of such graphs. Instead of the elimination order being of the vertices of the
given graph G as in the one cop case, it is an ordering of the vertices of the (k + 1)-fold
categorical product of G with itself. Most of our results hold for variations of the game and
some of them extend to infinite graphs.
11071
40
11071 – Theory and Applications of Graph Searching Problems (GRASTA 2011)
3.17
Some thoughts on constrained cops-and-robbers
Gena Hahn (Université de Montréal, CA)
License
Creative Commons BY-NC-ND 3.0 Unported license
© Gena Hahn
This talk is essentially about open questions. First we propose a more general setting for
cops-and-robbers games on graphs. Next, we suggest a way to model position and move
constraints for the games and observe that there is a partially ordered set of constraints.
We then ask what the structure of the poset might be, having observed that the theorem
of Nowakowski and Winkler that characterizes cop-win graphs via a binary relation on the
set of vertices carries over to the general setting. We close by suggesting that graphs that
have some, but not all, loops should be studied, as well as tournaments, and propose a few
problems.
3.18
Hypergraph searching as notion justification
Andrei Krokhin (University of Durham, GB)
License
Creative Commons BY-NC-ND 3.0 Unported license
© Andrei Krokhin
We discuss a class of hypergraphs that appeared recently in the study of the constraint
satisfaction problem. We show that this class can be described by a natural variant of the
hypergraph searching game.
3.19
Monitoring on a Grid
Dieter Mische (UPC – Barcelona, ES)
License
Creative Commons BY-NC-ND 3.0 Unported license
© Dieter Mische
We consider a set of g walkers W moving on the n × n integer grid. Initially, each walker
chooses a vertex u.a.r., and in each step, each walker chooses u.a.r. and independently from
the other walkers, one neighboring vertex. Moreover, we are given a set D of fixed devices,
which are also placed on the integer grid. The devices are used to read data from walkers,
and a device can read data of a walker if the walker is within a certain grid distance. We
give bounds on the expected number of steps it takes to read data from all walkers for the
case where all devices are put onto the halving line of the grid and for the case where all
devices are regularly spread on the grid (in a grid-like way).
Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer, and Dimitrios M. Thilikos
3.20
41
LIFO-search
Dimitrios M. Thilikos (National and Kapodistrian University of Athens, GR)
License
Creative Commons BY-NC-ND 3.0 Unported license
© Dimitrios M. Thilikos
We study a variant of classic fugitive search game called LIFO-search where searchers are
ranked, i.e. are assigned different numbers. The additional rule is that a searcher can be
removed only if no searchers of lower rank are in the graph at that moment. We introduce the
notion of shelters in graphs and we prove a min-max theorem implying their equivalence with
the tree-depth parameter. As shelters provide escape strategies for the fugitive, this implies
the the LIFO-search game is monotone and that the LIFO-search parameter is equivalent
with the one of tree-depth.
4
4.1
Open problems
Cops and Robbers, parameterized algorithms
Fedor V. Fomin and Petr Golovach
License
Creative Commons BY-NC-ND 3.0 Unported license
© Fedor V. Fomin and Petr Golovach
By making use of backtracking algorithm, it is possible to decide if k cops can win on an
n-vertex graph in time nO(k) . It is easy to show that if the treewidth a graph is at most t,
then the cop number of G is at most t + 1. Thus on graphs of constant treewidth computing
the minimum number of cops sufficient to win can be done in polynomial time. What is
the parameterized complexity of the problem parameterized by the treewidth of the graph?
Similar questions can be asked about the parameterization by the clique-width, the genus,
and by the size of the excluded minor. The cop number of a graph is bounded by functions
of these parameters.
4.2
Computing edge and nodes search numbers on special graph
classes
Pinar Heggernes
License
Creative Commons BY-NC-ND 3.0 Unported license
© Pinar Heggernes
Let es(G) and ns(G) be the edge and node search numbers of a graph G, respectively. These
parameters are closely related: ns(G) − 1 ≤ es(G) ≤ ns(G) + 1. In general, both parameters
are NP-hard to compute, but there are families of graphs, like interval graphs, split graphs,
and cographs, on which both parameters can be computed in polynomial time. Is there there
a class F of graphs such that ns(G) can be computed in polynomial time for every graph
G ∈ F, whereas computing the edge search number is NP-hard on F? Natural candidate
classes to look at are those on which the computation of node search number, or equivalently
pathwidth, can be done in polynomial time, but no results are known on the computation of
their edge search number. As a first case to consider, ns(G) can be computed in polynomial
11071
42
11071 – Theory and Applications of Graph Searching Problems (GRASTA 2011)
(even linear) time if G is a permutation graph. Is the edge search number of permutation
graphs computable in polynomial time?
4.3
Connected node search number
Dimitrios M. Thilikos
License
Creative Commons BY-NC-ND 3.0 Unported license
© Dimitrios M. Thilikos
Let ns(G) and cns(G) be the node and connected node search numbers of graph G. We
denote by mcns(G), the monotone connected search number It was recently shown that
ns(G) ≤ cns(G) ≤ mcsn(G) ≤ 2 · ns(G).
Since deciding if ns(G) ≤ k is fixed parameter tractable parameterized by k, this gives an
FPT approximation algorithm for connected search number. Is deciding cns(G) ≤ k or
mcns(G) ≤ k in FPT?
It is known that numbers cns(G) and mcns(G) can be different. How much can they
be different? Is it correct that for almost all graphs cns(G)/mcns(G) → 1 as the size of G
goes to infinite?
It is believed that the parameter cns(G) is closed under contractions, i.e., contractions
of edges do not make the parameter increase. Is there a formal proof of this? Is deciding
cns(G) ≤ k in NP?
4.4
Span-width
Isolde Adler
License
Creative Commons BY-NC-ND 3.0 Unported license
© Isolde Adler
We say that a graph G has a span-width at most k, if there is a tree decomposition of G of
width k such that every vertex belongs to at most k + 1 bags. What is the parameterized
complexity of deciding if the span-width of a graph is at most k, parameterized by k? Similar
question for tree-spanners.
4.5
Cop number of toroidal graphs
Gena Hahn
License
Creative Commons BY-NC-ND 3.0 Unported license
© Gena Hahn
The long standing conjecture of Schroeder is that the cop number of a graph of genus g is at
most g + 3.
It is known that for toroidal graph this number is at most 4. Do toroidal graphs have
cop number at most 3 as conjectured by Andreae in 1986? Or is there a toroidal graph that
actually needs 4 cops?
Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer, and Dimitrios M. Thilikos
4.6
43
Kelly-width
Paul Hunter
License
Creative Commons BY-NC-ND 3.0 Unported license
© Paul Hunter
The Kelly-width of a digraph D is defined as the minimum number of searchers required
to catch an invisible, inert fugitive with a (fugitive-)monotone strategy. “Inert” means the
fugitive is unable to move from a vertex unless a searcher is about to land on that vertex.
When the fugitive is able to move, he may move along any directed path not occupied by a
searcher. For a more precise definition, see Hunter & Kreutzer [4]. Digraphs of Kelly-width
1 are precisely the acyclic digraphs, and there is a known polynomial time algorithm for
deciding if a digraph has Kelly-width at most 2, see [6].
Is deciding if the Kelly-width of digraph D is at most k in PTIME for any fixed k ≥ 3?
4.7
Ray searching
Spyros Angelopoulos
In the m-lane ray search problem we are given a set of m semi-infinite lanes with a common
origin O. A target is placed at an unknown ray at distance d from the origin. We seek a
strategy that minimizes the worst-case ratio cost/d, where cost denotes the overall distance
traversed by the searcher up to the point it locates the target.
This problem has been solved in its deterministic variant by Gal [3]. The question
of finding randomized strategies that minimize the worst-case ratio E[cost]/d is not quite
settled. In [5] a randomized strategy is presented which however is optimal only in the class
of round-robin strategies. Can we find optimal randomized strategies without any restrictive
assumptions?
4.8
Ratio of monotonicity
Stephan Kreutzer
License
Creative Commons BY-NC-ND 3.0 Unported license
© Stephan Kreutzer
Consider two version of searching on a directed graph:
Inert invisible fugitive game.
Visible fugitive game.
Both problems are known to be non-monotone. Is there a number d such that the ratio
between monotone and non-monotone versions of these games is at most d? More generally,
is there an FPT approximation of non-monotone via monotone parameters?
11071
44
11071 – Theory and Applications of Graph Searching Problems (GRASTA 2011)
4.9
Best strategy to catch the drunk robber
Dimitrios M. Thilikos
License
Creative Commons BY-NC-ND 3.0 Unported license
© Dimitrios M. Thilikos
So far, in all cop and robber game settings, cops where considered to be omniscient and lucky
in the sense that they will always take the best decision in order to avoid, or delay, capture.
An interesting topic would be to study the setting where the robber moves randomly and
the cops are clever. This induces a mix between classic graph searching and random walks.
An example is given below:
Given a graph such as a line, a cycle, or a (torodial) grid, or a 3-regular graph, consider a
robber that chooses randomly its first position and then moves randomly in neighbor nodes
of the graph. Assume also that there are so many cops as the searching number of the graph.
The cops play first, may move simultaneously. The two parts play in rounds. The objective
here is to compute the minimum, over all cop strategies, expected time of arrest of the the
(drunk) robber.
Question 1. Are all the optimal strategies monotone in the sense that the expected
capture time does not change if cop strategies are restricted to those that do not visit again
an already searched location? (The question has some meaning even when the number of
cops is smaller than the search number of the graph.)
Question 2. How the expected capture time changes when there are less cops than the
cop number?
Question 3. What is the ratio between the expected capture time for a drunk robber
and the maximum capture time a “sober” robber (i.e., one that makes its best to avoid
capture). Is this ratio common for many (or even all) graphs? Is it a constant such as 2?
4.10
Escaping from random cops
Pierre Fraigniaud
License
Creative Commons BY-NC-ND 3.0 Unported license
© Pierre Fraigniaud
The paper [1] analyzes a randomized cop-and-robber game on graphs. The cop and the
robber do not see each other, unless they are on the same node, in which case the robber
is caught. They both move along the edges of the graph, one edge per round, playing in
turn. Given a randomized cop strategy, the escape length for that strategy is the worst case
expected number of rounds it takes the cop to catch the robber, where the worst case is
with regards to all (possibly randomized) robber strategies. Adler et al. [1] proposes a cop
strategy with an escape length of O(n log D) in n-node diameter-D graphs. On the other
hand, there is a trivial Ω(n) lower bound on the escape length.
Open problem: close the gap between the two bounds.
One restricted case that may deserve attention is the case where the cop is bounded to
apply simple random walk. In that case, is the best strategy for the robber the one consisting
in placing itself at the node with lowest steady state probability, and stay idle? Or, if the
initial position of the cop given, is the best strategy for the robber the one consisting in
placing itself at the node with highest hitting time, and stay idle?
Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer, and Dimitrios M. Thilikos
45
References
1 M. Adler, H. Räcke, N. Sivadasan, C. Sohler, B. Vöcking, Randomized PursuitEvasion in Graphs, ICALP 2002: 901-912
2 T. Andreae, On a pursuit game played on graphs for which a minor is excluded, J. Combin.
Theory Ser. B, 41 (1986), pp. 37–47.
3 S. Gal, Minimax solutions for linear search problems, SIAM J. on Applied Math., 27
(1974), pp. 17–30.
4 P. Hunter and S. Kreutzer, Digraph measures: Kelly decompositions, games, and
orderings. Theor. Comput. Sci. 399(3): 206-219 (2008)
5 M-Y. Kao and Y. Ma and M. Sipser and Y.L. Yin, Optimal constructions of hybrid
algorithms, Journal of Algorithms, 29 (1998), pp. 142–164.
6 D. Meister, J. A. Telle, M. Vatshelle, Recognizing digraphs of Kelly-width 2. Discrete
Applied Mathematics 158(7): 741-746 (2010)
11071
46
11071 – Theory and Applications of Graph Searching Problems (GRASTA 2011)
Participants
Isolde Adler
Univ. Frankfurt am Main, DE
Gena Hahn
Université de Montréal, CA
Carme Alvarez
UPC – Barcelona, ES
Pinar Heggernes
University of Bergen, NO
Spyros Angelopoulos
CNRS - Paris, FR
Paul Hunter
University of Oxford, GB
Dietmar Berwanger
ENS – Cachan, FR
David Ilcinkas
Université Bordeaux, FR
Lélia Blin
Université d’Evry, FR
Marcin Jurdzinski
University of Warwick, GB
Nancy Clarke
Acadia Univ. – Wolfville, CA
Marcin Kaminski
University of Brussels, BE
Dariusz Dereniowski
Gdansk Univ. of Technology, PL
Athanasios Kehagias
Aristotle University of
Thessaloniki, GR
Josep Diaz
UPC – Barcelona, ES
Amalia Duch Brown
UPC – Barcelona, ES
Fedor V. Fomin
University of Bergen, NO
Stephan Kreutzer
University of Oxford, GB
Andrei Krokhin
University of Durham, GB
Pierre Fraigniaud
Univ. Paris-Diderot, CNRS, FR
Margaret-Ellen Messinger
Mount Allison University –
Sackville, CA
Petr Golovach
University of Durham, GB
Zoltan Miklos
EPFL – Lausanne, CH
Dieter Mitsche
UPC – Barcelona, ES
Nicolas Nisse
INRIA Sophia Antipolis, FR
Jan Obdrzálek
Masaryk University, CZ
Xavier Perez Gimenez
MPI für Informatik –
Saarbrücken, DE
Pawel Pralat
West Virginia Univ. –
Morgantown, US
Maria Serna
UPC – Barcelona, ES
Dimitrios M. Thilikos
National and Kapodistrian
University of Athens, GR
Tomas Valla
Charles University – Prague, CZ
Erik Jan van Leeuwen
University of Bergen, NO
Peter Widmayer
ETH Zürich, CH
Boting Yang
University of Regina, CA